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Some extensions of the uncertainty principle

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  • Zozor, Steeve
  • Portesi, Mariela
  • Vignat, Christophe

Abstract

We study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [I. Bialynicki-Birula, Formulation of the uncertainty relations in terms of the Rényi entropies, Physical Review A 74 (5) (2006) 052101] and Zozor et al. [S. Zozor, C. Vignat, On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles, Physica A 375 (2) (2007) 499–517]. Those inequalities can be considered as generalizations of the Heisenberg uncertainty principle, since they measure the mutual uncertainty of a wave function and its Fourier transform through their associated Rényi entropies with conjugated indices. We consider here the general case where the entropic indices are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indices α and β in the plane (α,β). Our results explain and extend a recent study by Luis [A. Luis, Quantum properties of exponential states, Physical Review A 75 (2007) 052115], where states with quantum fluctuations below the Gaussian case are discussed at the single point (2,2).

Suggested Citation

  • Zozor, Steeve & Portesi, Mariela & Vignat, Christophe, 2008. "Some extensions of the uncertainty principle," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(19), pages 4800-4808.
  • Handle: RePEc:eee:phsmap:v:387:y:2008:i:19:p:4800-4808
    DOI: 10.1016/j.physa.2008.04.010
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    References listed on IDEAS

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    1. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(4), pages 691-705, August.
    2. Portesi, M & Plastino, A, 1996. "Generalized entropy as a measure of quantum uncertainty," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 225(3), pages 412-430.
    3. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(5), pages 879-883, October.
    4. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(6), pages 1195-1198, December.
    5. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(1), pages 225-228, February.
    6. Zozor, S. & Vignat, C., 2007. "On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 375(2), pages 499-517.
    7. Vignat, C & Hero III, A.O & Costa, J.A, 2004. "About closedness by convolution of the Tsallis maximizers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 147-152.
    8. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(2), pages 411-413, April.
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    1. de Moraes, Celso Francisco & Silva, Messias Borges, 2018. "Framework for conformity assessment based on an analogy with the Uncertainty Principle of Quantum Mechanics (FCAUP)," International Journal of Production Economics, Elsevier, vol. 203(C), pages 394-403.

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